Figure REF  compares the quality of estimated \(\mu \)  between our exact two-arm method (using published / known reference sample inclusion probabilities) with the pseudo likelihood methods. The first row presents the combined reference and convenience sample using the true values for the latent convenience sample weights as a comparator for all methods. The second row presents the combined reference and convenience samples now using the estimated convenience sample inclusion weights under our two-arm method. The performance is very similar to using the true inclusion weights for the convenience sample. The third row presents the LCW method of [1]}, which performs relatively poorly due to high estimation variation expressed by this method in estimation of \(\pi _{cmi}\)  for our moderate sample sizes \((n_r,n_c) = (400,\sim 800)\) . We achieve best performance for high-overlap convenience samples (labeled (H)). The last row presents the same, but using the WVL pseudo likelihood method of [2]} that expresses less variation in estimation of \(\pi _{cmi}\)  than does LCW (though still substantially higher than our two-arm method). Yet, even though the estimated weights under WVL are biased under both low- and high-overlap samples, the method performs similarly in estimation of \(\mu \)  to our two-arm method because the bias for WVL is largest at high values for \(\pi _{ci}\)  while most sampled units are assigned \(\pi _{ci} < 0.75\) . The low-overlap samples produce notably worse coverage under WVL due to the bias and failure to account for uncertainty in \(\pi _{rmi}\)  by using them as plug-in.
